The least Euclidean distortion constant of a distance-regular graph

TL;DR

This paper investigates the least Euclidean distortion of distance-regular graphs, confirming Vallentin's conjecture for some families, providing counterexamples, and proposing alternative conjectures with proofs.

Contribution

It confirms Vallentin's conjecture for certain distance-regular graphs, presents counterexamples, and introduces and proves three new conjectures.

Findings

Vallentin's conjecture holds for several families of distance-regular graphs.

Counterexamples show the maximum contraction can occur at distance d-1.

Three alternative conjectures are proposed and proven for specific graph families.

Abstract

In 2008, Vallentin made a conjecture involving the least distortion of an embedding of a distance-regular graph into Euclidean space. Vallentin's conjecture implies that for a least distortion Euclidean embedding of a distance-regular graph of diameter $d$, the most contracted pairs of vertices are those at distance $d$. In this paper, we confirm Vallentin's conjecture for several families of distance-regular graphs. We also provide counterexamples to this conjecture, where the largest contraction occurs between pairs of vertices at distance $d{-}1$. We suggest three alternative conjectures and prove them for several families of distance-regular graphs.